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The use of Gaussian orbitals in electronic structure theory (instead of the more physical Slater-type orbitals) was first proposed by Boys[2] in 1950. The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4—5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.

For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the cartesian basis, and the spherical functions can be simply expressed using the cartesian functions.[3]

where Ylm(θ,ϕ){\displaystyle Y_{lm}(\theta ,\phi )} is a spherical harmonic, l{\displaystyle l} and m{\displaystyle m} are the angular momentum and its z{\displaystyle z} component, and r,θ,ϕ{\displaystyle r,\theta ,\phi } are spherical coordinates.

where cp{\displaystyle c_{p}} is the contraction coefficient for the primitive with exponent αp{\displaystyle \alpha _{p}}. The coefficients are given with respect to normalized primitives, because coefficients for unnormalized primitives would differ by many orders of magnitude. The exponents are reported in atomic units. There is a large library of published Gaussian basis sets optimized for a variety of criteria available at the EMSL basis set exchange.

Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the Gaussian basis.[4] Since then much work has been done to speed up the evaluation of these integrals which are the slowest part of many quantum chemical calculations. Živković and Maksić (1968) suggested using Hermite Gaussian functions,[5] as this simplifies the equations. McMurchie and Davidson (1978) introduced recursion relations,[6] which greatly reduces the amount of calculations. Pople and Hehre (1978) developed a local coordinate method.[7] Obara and Saika introduced efficient recursion relations in 1985,[8] which was followed by the development of other important recurrence relations. Gill and Pople (1990) introduced a 'PRISM' algorithm which allowed efficient use of 20 different calculation paths.[9]

The POLYATOM System[10] was the first package for ab initio calculations using Gaussian orbitals that was applied to a wide variety of molecules.[11] It was developed in Slater's Solid State and Molecular Theory Group (SSMTG) at MIT using the resources of the Cooperative Computing Laboratory. The mathematical infrastructure and operational software were developed by Imre Csizmadia,[12] Malcolm Harrison,[13] Jules Moskowitz[14] and Brian Sutcliffe.[15]